Find the power series representation for the function f(x)=x/4−x.

Find the power series representation for the function f(x)=x/4−x. Correct Answer ∞∑n=0xn+1/4n+1

So, again, we’ve got an x in the numerator. f(x)=x*1/4−x. If there is a power series representation for g(x)=1/4−x, there will be a power series representation for f(x). Suppose, g(x)=1/4*1/1−x4. To get a power series representation is to replace the x with x4. Doing this gives, g(x)=1/4 ∞∑n=0 xn/4n (xn/4 nprovided ∣x/4∣<1) ⇒ g(x) = 1/4 ∞∑n=0 xn/4n = ∞∑n=0 xn/4n+1. The interval of convergence for this series is, ∣x/4∣<1⇒1/4 |x|<1⇒|x|<4. Now, multiply g(x) by x and we have f(x)=x*1/4−x=x ⇒ ∞∑n=0 xn/4n+1 = ∞∑n=0xn+1/4n+1 and the interval of convergence will be |x|<4.